The Plancherel formula, the Plancherel theorem, and the Fourier transform of orbital integrals

Herb, Rebecca A.; Sally, Jr Paul J.; Sally, Paul

doi:10.1090/conm/557/11023

“…Although the Fourier Transform gives complex results in general, the DFT can be represented by d real numbers for real input data of dimension d. Importantly, these real numbers can be bounded. Given ⃗ y, we have ∥⃗ y ∥ 2 = ∥DFT(⃗ y )∥ 2 (Plancherel Theorem [30]). So if we ensure that our vectors are normalized so that ∥⃗…”

Section: A Discrete Fourier Transformmentioning

confidence: 99%

Aggregation and Transformation of Vector-Valued Messages in the Shuffle Model of Differential Privacy

Scott

¹

,

Cormode

²

,

Maple

³

2022

IEEE Trans.Inform.Forensic Secur.

5

0

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Advances in communications, storage and computational technology allow significant quantities of data to be collected and processed by distributed devices. Combining the information from these endpoints can realize significant societal benefit but presents challenges in protecting the privacy of individuals, especially important in an increasingly regulated world. Differential privacy (DP) is a technique that provides a rigorous and provable privacy guarantee for aggregation and release. The Shuffle Model for DP has been introduced to overcome challenges regarding the accuracy of local-DP algorithms and the privacy risks of central-DP. In this work we introduce a new protocol for vector aggregation in the context of the Shuffle Model. The aim of this paper is twofold; first, we provide a single message protocol for the summation of real vectors in the Shuffle Model, using advanced composition results. Secondly, we provide an improvement on the bound on the error achieved through using this protocol through the implementation of a Discrete Fourier Transform, thereby minimizing the initial error at the expense of the loss in accuracy through the transformation itself. This work will further the exploration of more sophisticated structures such as matrices and higher-dimensional tensors in this context, both of which are reliant on the functionality of the vector case.

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“…Although the Fourier Transform gives complex results in general, the DFT can be represented by d real numbers for real input data of dimension d. Importantly, these real numbers can be bounded. Given y, we have y 2 = DFT( y) 2 (Plancherel Theorem [30]). So if we ensure that our vectors are normalized so that x i 1 = 1, then DFT( x i ) 2 = x i 2 ≤ x i 1 .…”

Section: Discrete Fourier Transformmentioning

confidence: 99%

Aggregation and Transformation of Vector-Valued Messages in the Shuffle Model of Differential Privacy

Scott,

Cormode,

Maple

2022

Preprint

0

View full text Add to dashboard Cite

Advances in communications, storage and computational technology allow significant quantities of data to be collected and processed by distributed devices. Combining the information from these endpoints can realize significant societal benefit but presents challenges in protecting the privacy of individuals, especially important in an increasingly regulated world. Differential privacy (DP) is a technique that provides a rigorous and provable privacy guarantee for aggregation and release. The Shuffle Model for DP has been introduced to overcome challenges regarding the accuracy of local-DP algorithms and the privacy risks of central-DP. In this work we introduce a new protocol for vector aggregation in the context of the Shuffle Model. The aim of this paper is twofold; first, we provide a single message protocol for the summation of real vectors in the Shuffle Model, using advanced composition results. Secondly, we provide an improvement on the bound on the error achieved through using this protocol through the implementation of a Discrete Fourier Transform, thereby minimizing the initial error at the expense of the loss in accuracy through the transformation itself. This work will further the exploration of more sophisticated structures such as matrices and higher-dimensional tensors in this context, both of which are reliant on the functionality of the vector case.

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The Space $$\mathcal{P}_{n}$$ of Positive n × n Matrices

Terras

¹

2016

Harmonic Analysis on Symmetric Spaces—Higher Rank Spaces, Positive Definite Matrix Space and Generalizations

0

1

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The Plancherel formula, the Plancherel theorem, and the Fourier transform of orbital integrals

Abstract: Abstract. We discuss various forms of the Plancherel Formula and the Plancherel Theorem on reductive groups over local fields.

Cited by 3 publications

References 94 publications

Aggregation and Transformation of Vector-Valued Messages in the Shuffle Model of Differential Privacy

Aggregation and Transformation of Vector-Valued Messages in the Shuffle Model of Differential Privacy

Aggregation and Transformation of Vector-Valued Messages in the Shuffle Model of Differential Privacy

The Space $$\mathcal{P}_{n}$$ of Positive n × n Matrices

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